Modelling and simulation of a feed drive using the transmission line modelling technique

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Transactions on Engineering Sciences vo 44, 2003 WT Press, www.witpress.com, SSN 1743-3533 Modeing and simuation of a feed drive using the transmission ine modeing technique V.Y. Moreno-Castaneda, C.

1 Transactions on Engineering Sciences vo 44, 2003 WT Press, SSN Modeing and simuation of a feed drive using the transmission ine modeing technique V.Y. Moreno-Castaneda, C. Pisaru, J.M. Freeman & D.G. Ford Utra Precision Engineering Centre, University of Huddersfied, UK Abstract The paper focuses on the modeing and simuation of a test rig and a digita drive is based on the transmission ine modeing technique. The rig is a positioning system equivaent to a CNC machine too feed drive. Accurate representations of dynamic systems incuded in eectromagnetic, power eectronic and some mechanica appications have been obtained unti now by appying the transmission ine modeing technique. The paper proposes a nove appication of th~s method for simuation of cosed oop contro of the position of a test rig. The mode incudes the distributed parameter dynamics of the feed drive and the effect of mass distribution. This is based on the fexibiity of transmission ine modeing to deveop umped and distributed parameter systems. Aso a transmission ine mode for the controer integrated into the digita drive is deveoped. Encouraging simuation resuts are obtained by using MATLAB. n this way, the CNC machine toos feed drives coud be modeed with increasing accuracy as hybrid, distributed-umped representations using numerica methods. 1 ntroduction Precise modes refecting the behaviour of non-inear dynamic systems (such as CNC machine toos) become increasingy important when high-performance contro is required. Lumped parameter modes [] with the oad inertia referred to the motor shaft have described the operation of CNC machine toos feed drives for many years. However, they did not refect the fu dynamic behaviour of the machine toos.

Transactions on Engineering Sciences vo 44, 2003 WT Press, www.witpress.

2 Transactions on Engineering Sciences vo 44, 2003 WT Press, SSN Laser Metroogy und Muchiize Performance V The next step was the appication of a moduar approach [2] where the effect of mechanica transmission upon the eectrica drive is considered as a perturbing oad torque. The simuated resuts did not compare we with the measured ones so a hybrid mode containing connected stiffness, inertia and expicit damping coefficients was deveoped [3]. The ba-screw was modeed using seven modues and ba-screw fexibiity was apportioned between each. Athough it produced simiar dynamic responses to the same stimui as the machine too feed drive system, ony an anaogue drive was modeed and the stiffness cacuations were made for ony one position of the worktabe. Sti the mode-system correspondence has to be improved in order to refect the point wise and the distributed features of the CNC machine toos feed drives. The transmission ine (TL) modeing technique is chosen to deveop a mode which represents the dynamic behaviour of feed drives. Reativey simpe procedures are empoyed by assuming that capacitors, inductors and resistances can represent the eements of hybrid systems such as feed drives. Another reason for choosing this numerica modeing technique is the speed of processing. The deveoped mode is intended to become a part of on-ine condition-monitoring methods. Therefore very fast and accurate simuated resuts wi be necessary to be compared with direct responses from the machme. The method of approach was firsty vaidated against the anaogue transform technique proposed by Bartett [4]. The suggested hybrid mode for a rotor she has been impemented in MATLAB and the simuated transient resuts compared we with those presented by Bartett. The effect of distributed mass / inertia and stiffness was very evident [4, 51. The paper presents the deveopment of a transmission ine mode for a test rig existing at the University of Huddersfied. The foowing sections contain a brief description of the TL method and the process of buiding TL modes for the components of the digita drive. The differentia equations describing their dynamic behaviour and the corresponding TL modes are anaysed for every component of the feed drive. The TL mode for the digita controer represents an important contribution. This TL mode incuding the hybrid nature of the feed drive and the mass dispacement is impemented in MATLAB. Simuation resuts are the same as cacuated using cassica contro theory and due consideration to machme too eements. 2 Transmission ine method TLM is a numeric method used to sove time-dependent or transient probems. The method approximates continuous space representing a system as a mesh of transmission ines due to the mathematica equivaence between the system equations and the equations for votages and currents on the TL mesh. Two equivaent circuits are used in the TLM technique: the stub and the ink circuits. TLM inks are two-port one-dimensiona buiding bocks that can be used for one-, two- or three-dimensiona modeing. On the other hand, TLM stubs are one-port units, which can be used for soving circuits and equations, and are used to compement TLM inks.

3 Transactions on Engineering Sciences vo 44, 2003 WT Press, SSN Laser Metroogy and Machine Performance V 195 Using the theory stated in [6], any eectrica circuit can be represented as a network of transmission sections by simpy repacing the reactive components with corresponding stubs. Variabes such as votage and current are regarded as discrete puses bouncing to and from the nodes of these stubs at each time step. The votage and current in each component is determined from the incident (e') and refected (e') puses in a port (Figure 1). The TLM operation begins with the incident puses representing the initia conditions being injected into the network. ncident puses take a time (At) to trave between ports. When incident puses reach a port (nodes) refected puses are generated according to boundary conditions. The refected puses thus become the incident puses in the next time step. On incidence to the node, the puse wi interact with other parts of the circuit. :: a) TLM stub Figure 1 : TLM Units f kei is known at time step k, the votages and current in Figure 1 may be cacuated. Taking g as the discrete stimuus appied to the TL, gives:,i = (,e-2,ei)/~,, (1) From the equivaent branch,e = 2,e' +,iz, (2) i the refected puse wi be,er = ke- ke (3) the refected puse becomes the next incident puse, hence i e = r, er (4) Now, with k +~e obtained from eqn (4),,+Ji may be obtained from eqn (1). Then, the process, known as the scattering agorithm, is repeated for as ong as desired. Depending on the nature of the eement to represent, Zo and Tcan be chosen accordingy. 3 Test rig mode using the transmission ine method n Figure 2, a bock diagram of the studied test rig is shown. The mode of the system is buit firsty by deveoping the equations that characterize the behaviour of each component, and after that by representing them with their corresponding TLM equivaent.

4 Transactions on Engineering Sciences vo 44, 2003 WT Press, SSN Laser Metroogy und Muchiize Performance V Coniroer Direct Screw N~t and couping shaf oad t t 3.1 Digita axis - controer mode Figure 2: Bock diagram of the test rig The drive controer is impemented with an agorithm foowing the principe of cascade contro [7]. Thus, the motor is driven by an inverter by means of puse width moduated (PWM) signas , Controer nverter Machine! 1,nterface PWM Actua cunenf Actua speed 1 Actua poran Figure 3: Bock diagram of controer Set point generator For a desired movement, the set point generator cacuates a veocity profie from the feed rate, maximum acceeration, and jerk. The veocity profie (v(tjj is then used to generate a set of set points, and times to be output. f the system has mutipe axes, the motions of individua axes are coordinated using an interpoation technique. For inear movements a inear interpoation is performed. This interpoation is achieved resoving the distance to move each axis during the cyce time (At,). For a two-axis movement (X, Y) the inear interpoation agorithm takes the form: RG near wth encoder d= Jx2 +y2 (6) Thus, a nomina position vaue (SJ is derived every 3ms from the previous nomina position (Sxo), the cacuated veocity (v), the distance the axis is going to move (X), the ength of the movement (4, and the cyce time. Simiary, the process is made simutaneousy for the y-axis using Y/D giving a nomina position vaue S,. Position controer The nomina position vaue (S,,,J is compared with the actua position vaue (S,,,J and the servo ag (SJ is cacuated (S,,=S,,,-S,,J. The ag is mutipied by a

5 Transactions on Engineering Sciences vo 44, 2003 WT Press, SSN Laser Metroogy and Machine Performance V 197 K, factor and passed on as a veocity vaue. This vaue is then added to the veocity feed-forward vaue (Vfdt=d(S,,,.J) to form the nomina veocity vaue for the veocity controer. Vnom =So *Kv +Vf (7) A TL stub modes the differentia term, then eqn (7) becomes Ymnt = ('non7 - Sac, ) Kv + Snonz'f + 2'; (8) For a simuation time At,, Zf = (at, 12) Speed controer The actua rotationa speed vaue is measured with a rotary encoder directy at the motor. The speed controer evauates the difference between the nomina and actua speed vaue (Ve=Vnom-V,,,). t outputs the nomina current vaue with a cyce time of 600 p (AtJ. A proportiona-integra-differentia strategy is used. The acceeration feed-forward is used in parae with the speed controer in order to minimize the spikes caused by changes in veocity. f P,, D and D are the respective proportiona factor, integra factor, differentia factor, and differentia factor for acceeration feed forward, the nomina current can be cacuated by Consequenty, the TLM representation is,,,, =ZAjiVji +2eji +PV,+Z,Ve+2e; +ZDV,+2eb (10) where ZAti = ~,(~t,/2), Z, =1(~t,12),and ZAf=D/(~t,/2). Current controer The current controer generates the reference current commands on the principe that the resutant stator current vector is aways kept in quadrature with the rotor fux vector. This is accompished using the current hysteresis controed PWM votage source inverter (VS) strategy. Detais of this approach and the mode for the power stage (inverter) can be found in [g]. 3.2 Permanent magnet synchronous motor mode n order to mode the motor, the machine performance equations are simpified to its vefevence frame representation according to [g]. This representation is a two-axes transformation of the stator and rotor quantities resuting in a set of equations, which possess constant coefficients. t aso has the advantage of keeping the magnitude of variabes in the same range. d L-i, =v, - Ri, - Lw,i, -k,w, dt Where, R and L represent the resistance and inductance of the windings of the motor. K, and kt are the eectric and torque constants of the motor, and W, the

6 Transactions on Engineering Sciences vo 44, 2003 WT Press, SSN Laser Metroogy und Muchiize Performance V anguar veocity of the motor. Subscripts d and q are used to identify the direct and quadrature currentsivotages respectivey. Because the motor is buit with a uniform air gap (round rotor) the generated torque, T,, is defined as: T =k *i a T q (13) Repacing the differentia components of eqn (5) and (6) by their correspondent TLM stub gives Z,i, + 2ej = v(, -Rid + Lw,i, (14) Zei, + 2eL = vq - Riq - Lweic - k,~, (15) Z, = L (At,, 12) (16) Mechanica dynamics of the motor A schematic representation of the mechanica components of the motor and its TLM mode is represented as shown in Figure 4. Figure 4: TLM mode of the mechanica dynamics of the motor J,, is the moment of inertia, b and b, represent the coefficient of friction at the bearings, and T, is the oad torque. The eectro- mechanica equation is An inductive TL stub represents the moment of inertia. Then, for the given propagation time (Ats), its characteristic impedance is z,, = J,,, /(At, 12). Thus, T, = (b, +b,)w, +Z,,w, +2ejn +T, (18) Direct couping For this stage of the modeing, the couping is considered as a gain kcc= 3.3 Dynamic mode of the ba-screw Nut and oad mode The rotationa torque (Td required to counterbaance the externa oad and the preoading of the ba screw is cacuated in concordance with [10]. T, = T,, + T, (19) Where Te1 is the torque due to externa oad and T, the torque due to preoading of the ba screw. The rotationa torque required to counterbaance the externa (T,J oad is

7 Transactions on Engineering Sciences vo 44, 2003 WT Press, SSN Laser Metroogy and Machine Performance V 199 Where Kb is the force to torque conversion constant and F the equivaent force on the direction of movement. f c the ead of the ba screw, n the reduction ratio, ef the efficiency of the ba screw, Kh = (d * n) 1 (2n" ef) (21) FCJf c -F;v,x wd,x'+ #'? Figure 5: Forces at the contact point between nut and screw shaft Hence, the foowing equations are vaid for the transformation of rotary movement to inear movement in the k' direction Then, the dynamics of the tabe are given by: F=Fc+Ff,+mac (23) Where, F, is the cutting force, Ff the frictiona force. Using eqn (23) into (20), d eqn(9) becomes Td = k,(~,, +~,,)+mk,?-wd +T, (24) dt A ba-screw is preoaded to make its axia cearance zero and reduce the dispacement with respect to the axia oad. Generay this torque can be expressed by T, = T,n [N-m] (25) T = O.OOS(tan P)-~.~F,,,,, / 2n (26) tan P =,[ (?i.bcd) (27) Where T, is the reference torque, P is the ead ange, F,, is the preoading oad, and BCD is the ba circe diameter. f K = k,e, 5 = k,ff, + T,, and J, = mk, eqn (24) becomes Repacing the differentia term by its TLM representation, gives r;i =T, +T, +Z,w, +2e: (29) z,,, = J, /(ats/2) (30) Figure 6: TLM mode of the effect of the tabe and nut on the screw shaft

8 Transactions on Engineering Sciences vo 44, 2003 WT Press, SSN Laser Metroogy und Muchiize Performance V The Screw shaft As the tabe is dispaced, the point of incidence of torque Td on the screw shaft changes over time. Consequenty, in order to anayse the behaviour of the system, the ength of the shaft is divided into j equa sections (Figure 7). Thus, the torque Td can be incuded in the mode jumping between boundaries of each section foowing the movement of the tabe. n this regard, it is considered that the difference of the effect of torque Td between the imits of each section is insignificant. Tfi and TP represent the torque due to friction at bearings. Figure 7: Screw shaft divided into j sections From [ ], the dynamic behaviour of a shaft subjected to torque about its ongitudina axis is represented by: This is the equation of a wave propagating at the veocity u =a, on a medium with characteristic impedance z,, = Jsm. G, Js, and p are the moduus of rigidity, moment of inertia and density of the shaft. Thus, each section can be modeed by a TLM ink as iustrated in Figure 8... Figure 8: TLM mode of the screw shaft Cj sections) Now the TL mode of the system may be deveoped by the interreation among the eements anaysed. n this view, torque Td represents a drop of effort at the interaction point between nut and screw. Accordingy, the TLM mode of this dynamic effect is iustrated in Figure 9. Then, the anguar veocities at each section are cacuated with w =(T, -T,, -za;)/z, (32) W, = (BA-, - A;) 1.2, for k=2,3 to j (33)

9 Transactions on Engineering Sciences vo 44, 2003 WT Press, SSN Laser Metroogy and Machine Performance V 201 wd = ( 2~: -q -2~; -T, -2~~+,)/(2~, +z,) (35) 4Jc section 2 -+p torque Td --/c section 3 --.F S S S 4 Simuated resuts Figure 9: Nut on section two The TL equations for the compete system are impemented in Matab environment. Simuation resuts for a desired feed rate of 4000 drnin are compared against experimenta resuts. Simuated and measured vaues for the reference current command and the actua speed of the motor are iustrated in Figures 10 and 11 respectivey. The sight difference between simuated and measured nomina current (Figure 10) can be attributed to the coefficients of the poynomia used to conform the speed profie. Resuts for speed (Figure 11) demonstrates the accuracy of the TLM mode in resembing the dynamic behaviour of the test rig. At this moment the backash is not incuded in the mode. t wi be incorporated in future modes to generate simuated resuts simiar to measured ones E 3000 'E E 2500 m m 500 $0 0s time [S] Figure 10: Nomina current 0 time [S] Figure 11: Actua speed 5 5 Concusions The modeing and simuation of the positioning system based on transmission ine modeing technique is an important contribution to machine too theory. The equations describing the dynamic behaviour of the test rig are converted into impedance and admittance forms enabing both continuous and discrete modes

Transactions on Engineering Sciences vo 44, 2003 WT Press, www.witpress.com, SSN 1743-3533 202 Laser Metroogy und Muchine Performance V to be accommodated.

10 Transactions on Engineering Sciences vo 44, 2003 WT Press, SSN Laser Metroogy und Muchine Performance V to be accommodated. This hybrid representation is impemented in MATLAB and the simuated resuts are in accordance with CNC machine too theory. There are severa steps, which have to be competed in order to estabish a systematic basis enabing the construction of hybrid modes for feed drives: a) incude geometric errors, and b) deveopment of a three-dimensiona mode for CNC machine too feed drives and vaidation against practica data. The described approach aows the easy construction of an accurate dynamic mode of feed drives. t represents the basis for future deveopment of a universa mathematica mode for modern CNC machine toos incorporating digita drives. Acknowedgements The authors woud ike to acknowedge that the main part of the work was carried out under EPSRC grant GR S for robust on-ine parameter identification appied to precision CNC machine toos. References [] Ford, D.G., A CAD-CAE Aid to Design of a Machine Too System, PhD thesis, The University of Huddersfied, UK, [2] Pisaru, C., Ford, D.G. & Freeman J. M,, A New Approach to the Modeing and Simuation of a CNC Machine Too Axis Drive. Proc. of the 4th nt. Con$ on Laser Metroogy and Machine Performance (LAMDAMAP 99), V. Chies & D. Jenkinson, WT Press: Newcaste, pp , [3] Pisaru, C., Parameter dentijkation and Hybrid Mathematica Modeing Techniques Appied to Non-inear Contro Systems, Ph.D. Thesis, The University of Huddersfied, UK, [4] Bartett, H., & Whaey, R., Anaogue soution to the modeing and simuation of distributed-umped parameter systems. Proceedings of MechE, 212(1), pp , [5] Moreno-Castaneda, V. Y., Transmission Line Modeing Technique Appied to Feed Drives, nterna report, University of Huddersfied, UK, [6] Christopouos, C. (1). The Transmission Line Modeing Method (TLM), The EEEOUP Series on Eectromagnetic Wave Theory: New York, [7] Heidenhain, TNC guide OEM, Version 2.1, [8] Low, T.S, Ding, Y.P. & Lock, K.S., Anaysis and contro of torque rippes in inverter-fed permanent-magnet synchronous motors. Proceedings of EE, 141(3), pp , [g] Prokop, L. & Grasbum, P., 3-Phase PM synchronous motor vector contro using DSP56F80x, Motoroa iterature distribution, AN1931/D, pp. 4-9, [01 THK, LM System- Ba Screws, Linear System Editoria Group: Japan, pp , [ ] Partridge, G.J, Christopouos, C. & Johns, P.B., Transmission ine modeing of shaft system dynamics, Proceedings of MechE, 210(C4), pp , 1987.

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